A uniform solid cylinder of mass $M = 3\  kg$ and radius $R = 10\  cm$ is connected about an axis through the cnetre of the cylinder to a horizontal spring with spring constant $8\ N/m$.The cylinder is pulled back, stretching the spring $1\,m$ from equilibrium.When released, the cylinder rolls without slipping. What is the speed of the center of th ecylinder when it returns to equilibrium? .................. $m/s$

A spring of force constant $800\, N/m$ has an extension of $5\,cm$. The work done in extending it from $ 5\,cm$ to $15 \,cm$ is ............. $\mathrm{J}$

$A$ ball of mass $m = 60gm$ is shot with speed $v_0 = 22m/s$ into the barrel of spring gun of mass $M = 240g$ initially at rest on $a$ frictionless surface. The ball sticks in the barrel at the point of maximum compression of the spring. What fraction of initial kinetic energy of the ball is now stored in the spring?

Two masses $m_1 = 2\,kg$ and $m_2 = 5\,kg$ are moving on a frictionless surface with velocities $10\,m/s$ and $3\,m/s$ respectively. An ideal spring is attached on the back of $m_2$ . The maximum compression of the spring will be ............... $\mathrm{m}$

Two particles with mass $m_1$ = $16\ kg$ and $m_2$ = $2\ kg$ slide as unit with a common velocity of $12\ ms^{-1}$ on a level frictionless surface. Between them is a compressed massless spring with spring constant $k$ = $100\ Nm^{-1}$ . The spring, originally compressed by $25\ cm$ , is suddenly released, sending the two masses, which are connected to the spring, flying apart from each other. The orientation of the spring w.r.t. the initial velocity is shown in diagram. What is the relative velocity of separation in $ms^{-1}$ , after the particles lose contact? ................$m/s$

Inside a lift, a spring (Force constant $k = 1000\ N/m$) and block ($mass = 1\  kg$) are both in a state of rest. Now the lift suddenly starts moving upwards with acceleration $a = g$. Find the maximum total compression in the spring in centimeter. ($g =10\ m/s^2$) :-